Consider the one dimensional Rosenau equation:
with initial condition
and the boundary conditions
where is a nonlinear term in of the type , here is a real
constant and is a positive integer.
The Rosenau equation is an example of a nonlinear partial differential equation, which governs the dynamics of dense discrete systems and models wave propagation in nonlinear dispersive media.
Recently, several numerical techniques like conforming finite element methods, mixed finite element methods, orthogonal cubic spline collocation methods, etc., were proposed to find the approximate solution of Rosenau equation. The different conforming finite element techniques are used to approximate the solution of Rosenau equation needs -interelement continuity condition and mixed finite element formulations are required -continuity condition. In this article discontinuous Galerkin finite element methods are used to approximate the solution.
The well-posedness results of (1.1)-(1.3) was proved by Park  and Atouani et al. in . Earlier, some numerical methods were proposed to solve the Rosenau equation (1.1)-(1.3) using finite difference methods by Chung , conservative difference schemes by Hu and Zheng  and Atouni and Omrani . Finite element Galerkin method was used by [2, 7], a second order splitting combined with orthogonal cubic spline collocation method was used by Manickam et al.  and Chung and Pani in  constructed a -conforming finite element method for the Rosenau equation (1.1)-(1.3) in two-space dimensions.
In recent years, there has been a growing interest in discontinuous Galerkin finite element methods because of their flexibility in approximating globally rough solutions and their potential for error control and mesh adaptation.
Recently, a cGdG method was proposed by Choo. et. al in . A subdomain finite element method using sextic b-spline was proposed by Battal and Turgut in . But constructing finite elements for fourth order problems becomes expensive and hence discontinuous Galerkin finite element methods can be used to solve fourth order problems .
In this paper, we introduce discontinuous Galerkin finite element methods (DGFEM) in space to solve the one dimensional Rosenau equation (1.1)-(1.3). Comparitive to existing methods our proposed method require less regularity.
The outline of the paper is as follows. In Section 2, we derive the discontinuous weak formulation of the Rosenau equation. In Section 3, we discuss the a priori bounds and optimal error estimates for the semidiscrete problem. In Section 4, we discretize the semidiscrete problem in the temporal direction using a backward Euler method and discuss the a priori bounds and optimal error estimates. Finally, in Section 5, we present some numerical results to validate the theoretical results.
Throughout this paper, denotes a generic positive constant which is independent of the discretization parameter which may have different values at different places.
2 Weak Formulation
and for . We denote this partition by consisting of sub-intervals . Below, we define the broken Sobolev space and corresponding norm
Now define the jump and average of across the nodes as follows. The jump of a function value across the inter-element node , shared by and denoted by and defined by
At the boundary and , we set
The average of a function value across the inter-element node , shared by and denoted by and defined by
At the boundary and , we set
We multiply (1.1) with and integrate over to obtain
Now, using integration by parts twice in (2.4), we arrive at
Summing over all the elements and using
Since is assumed to be sufficiently smooth, we have . Using this, we write as
We define the bilinear form as
Let and . Multiply (1.1) by and integrate
from to . Sum over all
and using (2.5), (2.6) and
(2.7), we obtain the weak formulation (2.9).
Conversely, let and , the space of infinitely differentiable functions with compact support in . Then, (2.9) becomes
Applying integration by parts twice on the second term on the left hand side of (2.11) to obtain,
as is compactly supported on . This immediately yields
Consider the node shared between and . Choose , multiply (2.12) by and integrate over to obtain
Applying integration by parts twice on the second term of (2.13) and using , we obtain
On the other hand, we have from (2.9) for the choice of and ,
Thus and hence, from (2.13), we obtain
This completes the proof. ∎
3 Semidiscrete DGFEM
In this section, we discuss the a priori bounds and optimal
error estimates for the semidiscrete Galerkin method.
We define a finite dimensional subspace of as
The weak formulation for the semidiscrete Galerkin method is to find such that
where is an appropriate approximation of which will be defined later.
3.1 A priori Bounds
In this sub-section, we derive the a priori bounds.
Define the energy norm
We note from  that is coercive with respect to the energy norm, i.e.,
for sufficiently large values of and .
Observe that (3.1
) yields a system of non-linear ordinary differential equations and the existence and uniqueness of the solution can be guaranteed locally using the Picard’s theorem. To obtain existence and uniqueness globally, we use continuation arguments and hence we need the followinga priori bounds.
Let be a solution to (3.1) and assume that is bounded. Then there exists a positive constant such that
On setting in (3.1), we obtain
We rewrite the equation (3.4) as
Integrating from to , we obtain
On using the coercivity of and the boundedness of , we arrive at
Using the Cauchy Schwarz inequality and the Poincaré inequality on the right hand side of (3.6), we obtain
An application of Gronwall’s inequality yields the desired a priori bound for . ∎
3.2 Error Estimates in the energy and -norm
In this subsection, we derive the optimal error estimates in energy
Often a direct comparison between and does not yield optimal rate of convergence. Therefore, there is a need to introduce an appropriate auxiliary or intermediate function so that the optimal estimate of is easy to obtain and the comparision between and yields a sharper estimate which leads to optimal rate of convergence for . In literature, Wheeler  for the first time introduced this technique in the context of parabolic problem. Following Wheeler , we introduce be an auxiliary projection of defined by
Now set the error and split as follows: , where and . Below, we state some error estimates for and its temporal derivative.
For and then there exists a positive constant independent of such that the following error estimates for hold:
We split as follows:
where , and
is an interpolant ofsatisfying good approximation properties. Now from (3.8), we have
We note that satisfies the following approximation property :
Set in (3.9) to obtain
A use of coercivity of and the assumption that is a sufficiently smooth interpolant of , we obtain
Now we estimate the first term as follows:
Estimating the second term using Hölder’s inequality, trace inequality and the Young’s inequality, we obtain
Similarly the last term can be estimated as
Now using , we obtain the energy norm estimate for . For the -estimate of , we use the Aubin Nitsché duality argument. Consider the dual problem
We note that satisfies the regularity condition . Consider
Since , we can write
where is a continuous interpolant of and satisfies the approximation property:
For the estimates of the temporal derivative of , we differentiate (3.8) with respect to and repeat the arguments. Hence, it completes the rest of the proof. ∎
The following Lemma is useful to prove the error estimates:
Let where . Then there exists a positive constant independent of such that,
We define the reference element as
Since , we have the following relation (refer ) for the norms in the reference element and the interval
In one space dimension, i.e., , we have
By the equivalence of norm (refer ), we have
Rearranging the terms and squaring on both sides, we obtain the desired estimate. ∎
Now we state and prove the following theorem.
Setting in (3.19), we obtain
Now, we write equation (3.20) as
Integrating with respect to from to and noting that , we obtain
We use integration by parts on the nonlinear term to obtain,
Using the Cauchy Schwarz’s and Young’s inequality, we bound the last term of (3.22) as
Now for the first term in (3.22), we use Hölder’s inequality to write
A similar bound for the second term can be obtained as follows. Using the Hölder’s inequality, we write
Using the trace inequality, we obtain